What Went Wrong in My Integration by Parts?

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SUMMARY

The discussion centers on the incorrect application of integration by parts to the integral of e^x/x. The user attempted to express the integral as a series but failed to recognize that the series does not converge. Participants confirmed that while the integration by parts method was applied correctly, the assumption that the resulting series converges is flawed, leading to an erroneous conclusion about the integral's value.

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Username007
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Here I used integration by parts to try to solve an integral (I got it wrong, it seems), I know this has no "simple" solution, but, can anyone explain me exactly what did I do wrong? Here is what I did:


\int\frac{e^x}{x}dx=\frac{e^x}{x}-(-1)\int\frac{e^x}{x^2}dx=\frac{e^x}{x}+1(\frac{e^x}{x^2}-(-2)\int\frac{e^x}{x^3}dx)=\frac{e^x}{x}+1!(\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\int\frac{e^x}{x^4}dx)=...=0!\frac{e^x}{x}+1!\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\frac{e^x}{x^4}+...

I just used integration by parts on the "remaining" integrals so I could produce a series, but the series is just...wrong. I would appreciate if you told me what is the error in my reasoning.
thanks.
 
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It doesn't, that's the point. Why can I use integration by parts and "say" that \int\frac{e^x}{x}dx equals a sum that does not converge?
 
And, thanks :)
 
Hi Username007! :smile:
Username007 said:
It doesn't, that's the point. Why can I use integration by parts and "say" that \int\frac{e^x}{x}dx equals a sum that does not converge?

integration by parts works fine so long as the ∑ is finite …

∫ ex/x dx

= ex(∑n=0k-1 n!/xn+1 - n! ∫ ex/xn+1 dx​

unfortunately, although we can usually rely on the final integral converging to zero as k -> ∞, in this case it doesn't! :rolleyes:
 
So, if the result of integration does not converge we can't establish the equality? (Thanks for you patience btw ^_^)
 
yes :smile:
 

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